Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

TL;DR

This paper introduces a generalized framework for dissipativity with dynamic supply rates, enabling stability analysis of nonlinear feedback systems through dissipation inequalities involving auxiliary systems, and unifies various existing stability criteria.

Contribution

It extends classical dissipativity theory to include dynamic supply rates and provides new stability conditions that encompass and unify several existing approaches.

Findings

Provides a Lyapunov-based stability analysis for nonlinear feedback systems.

Unifies multiple dissipativity-based stability criteria under a common framework.

Offers a simple coupling test for interconnected nonlinear systems.

Abstract

We propose a general notion of dissipativity with dynamic supply rates for nonlinear systems. This extends classical dissipativity with static supply rates and dynamic supply rates of miscellaneous quadratic forms. The main results of this paper concern Lyapunov and asymptotic stability analysis for nonlinear feedback dissipative systems that are characterised by dissipation inequalities with respect to compatible dynamic supply rates but involving possibly different and independent auxiliary systems. Importantly, dissipativity conditions guaranteeing stability of the state of the feedback systems, without concerns on the stability of the state of the auxiliary systems, are provided. The key results also specialise to a simple coupling test for the interconnection of two nonlinear systems described by dynamic (Psi, Pi, Upsilon, Omega)-dissipativity, and are shown to recover several existing results in the literature, including small-gain, passivity indices, static (Q, S, R)-dissipativity, dissipativity with terminal costs, etc. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made.